A composite function combines two functions so one acts on the result of the other; in fg(x), g acts first. Notes, worked examples and 10 questions answered.
Updated 14 Mar 2026 · Revise Right Team
Understanding Composite Functions is part of Composite functions in GCSE AQA Higher Further Maths.
3 min
Reading time
24
Lessons in this topic
GCSE AQA Higher
Pathway
Reviewed by Revise Right
Content review
What this lesson covers
A composite function combines two functions so one acts on the result of the other; in fg(x), g acts first. Notes, worked examples and 10 questions answered.
Define a composite function and explain what it means for two functions to act in succession
Apply the correct order of operations when evaluating fg(x), recognising that g acts first
Calculate composite function values by substituting step by step
Find an expression for a composite function such as fg(x) or gf(x) in simplified form
Key ideas to keep in view
Use this lesson to stay inside Composite functions while connecting the detail back to the wider Algebra topic.
Composite function: A function formed by applying two functions in succession. Used when the output of one function becomes the input of another.
fg(x): The composite function where g acts on x first, then f acts on the result. Read as 'f of g of x'.
Order of application: The rule that in fg(x), g is applied before f. The function written closest to x always acts first.
Revision notes
Read the core explanations from Understanding Composite Functions before testing yourself from memory.
1. What Is a Composite Function?
Imagine a vending machine that first doubles your money, then adds 50p. Two separate operations, one after the other — that chain of actions is exactly what a composite function is.
In maths, a composite function is what you get when the output of one function is fed directly into another. You are not doing anything new to the algebra — you are applying functions in a specific sequence.
The key insight is this: the function written closest to x always acts first. In $fg(x)$, that is $g$. In $gf(x)$, that is $f$. The letter on the outside acts last.
2. Evaluating a Composite Function at a Value
Before finding a general expression for $fg(x)$, it helps to practise with a specific number. This makes the two-stage process concrete and straightforward to check.
Suppose $f(x) = 2x + 1$ and $g(x) = x^2$. To find $fg(3)$, work from the inside out: find $g(3)$ first, then apply $f$ to that answer.
Worked example: Evaluating fg(3) Step by Step
Given $f(x) = 2x + 1$ and $g(x) = x^2$, find $fg(3)$.
$fg(3)$ means apply $g$ first, then $f$. — g is closest to the input 3, so it acts first. [M1 — correct order identified]
$g(3) = 3^2 = 3 \times 3 = 9$ — Substitute $x = 3$ into $g(x) = x^2$. The result is 9.
$f(9) = 2(9) + 1 = 18 + 1 = 19$ — Now substitute $x = 9$ into $f(x) = 2x + 1$. [A1 — correct final value]
Answer: $fg(3) = 19$
3. Finding a General Expression for fg(x)
Evaluating at a single number is useful for checking, but exam questions often ask you to find $fg(x)$ as an algebraic expression. The method is identical — substitute $g(x)$ wherever you see $x$ in $f$.
Worked example: Finding fg(x) as an Expression
Given $f(x) = 2x + 1$ and $g(x) = x^2$, find an expression for $fg(x)$.
$fg(x) = f(g(x))$ — g acts first, so replace the input of f with the entire expression g(x). [M1]
$fg(x) = f(x^2) = 2(x^2) + 1$ — Replace every x in $f(x) = 2x + 1$ with $x^2$.
$fg(x) = 2x^2 + 1$ — This is the final simplified expression. [A1]
Answer: $fg(x) = 2x^2 + 1$
Now compare that with $gf(x)$. Here $f$ acts first, so you substitute $f(x)$ into $g$. Watch how the result differs.
Worked example: Finding gf(x) as an Expression
Given $f(x) = 2x + 1$ and $g(x) = x^2$, find an expression for $gf(x)$.
$gf(x) = g(f(x))$ — f acts first this time, so replace the input of g with f(x). [M1]
$gf(x) = g(2x + 1) = (2x + 1)^2$ — Replace every x in $g(x) = x^2$ with $(2x + 1)$.
Notice that $fg(x) = 2x^2 + 1$ and $gf(x) = 4x^2 + 4x + 1$ — these are different expressions. Composite functions are not commutative: the order you apply them in changes the result.
4. Worked Exam-Style Questions
Exam questions on composite functions come in two main forms: evaluate at a number, or find a general expression. Both follow the same two-stage process. Here is a slightly more demanding example to build confidence with less familiar functions.
Worked example: Composite Function with a Rational Function
Given $f(x) = \dfrac{1}{x}$ and $g(x) = 3x - 2$, find $fg(x)$ and evaluate $fg(1)$.
$fg(x) = f(g(x)) = f(3x - 2)$ — g acts first, so substitute g(x) = 3x - 2 as the input to f. [M1]
$f(3x - 2) = \dfrac{1}{3x - 2}$ — In $f(x) = \frac{1}{x}$, replace x with $(3x - 2)$. [A1]
$fg(x) = \dfrac{1}{3x - 2}$ — This is the simplified composite expression.
$fg(1) = \dfrac{1}{3(1) - 2} = \dfrac{1}{3 - 2} = \dfrac{1}{1} = 1$ — Substitute x = 1 into the composite expression. [A1]
Answer: $fg(x) = \dfrac{1}{3x - 2}$, and $fg(1) = 1$
5. Practice Problems
The best way to lock in the method is to work through problems yourself before checking the answers. Try each one using the two-stage approach: identify which function acts first, apply it, then apply the second.
6. Exam Technique and Common Mistakes
Most marks lost on composite function questions come from one of two errors: applying the functions in the wrong order, or making an algebraic slip when substituting an expression (rather than a number) into a function.
Composite functions bring together function notation, substitution, and algebraic manipulation — three skills that appear across the paper. Mastering the order rule for $fg(x)$ means you can handle any combination of functions the exam presents.
Sources and content standards
Published by Revise Right. Use the official specification reference to check the requirements for your course. Content updated: 2026-03-14.
A composite function combines two functions f and g so that one acts on the result of the other. The notation fg(x) means apply g to x first, then apply f to that result: the function written closest to x always acts first.
How do you find fg(x)?
To find fg(x), substitute the whole expression g(x) wherever x appears in f. If f(x) = 2x + 1 and g(x) = x², then fg(x) = f(x²) = 2(x²) + 1 = 2x² + 1.
Is fg(x) the same as gf(x)?
Composite functions fg(x) and gf(x) are generally not equal, because composite functions are not commutative. With f(x) = 2x + 1 and g(x) = x², fg(x) = 2x² + 1, but gf(x) = (2x + 1)² = 4x² + 4x + 1.
How do you evaluate a composite function at a value?
To evaluate a composite function at a value, work from the inside out. For fg(3) with f(x) = 2x + 1 and g(x) = x², find g(3) = 3² = 9 first, then apply f to get f(9) = 2(9) + 1 = 19.
How do you find a composite function involving a fraction?
Substitute the inner function into the outer one in the same way. If f(x) = 1/x and g(x) = 3x − 2, then fg(x) = f(3x − 2) = 1/(3x − 2), and fg(1) = 1/(3 − 2) = 1.
Which function acts first in fg(x)?
In fg(x), the function g acts first, because the function written closest to x always acts first. The letter on the outside, f, acts last, so fg(x) means apply g to x, then apply f to the result.
How do you find gf(x)?
gf(x) is found by substituting f(x) into g, because f acts first. With f(x) = 2x + 1 and g(x) = x², gf(x) = g(2x + 1) = (2x + 1)², which expands to 4x² + 4x + 1.
How do you work out fg(3)?
fg(3) is worked out from the inside out: find g(3) first, then apply f to the answer. With f(x) = 2x + 1 and g(x) = x², g(3) = 9 and f(9) = 2(9) + 1 = 19, so fg(3) = 19.
What does f(g(x)) mean?
f(g(x)) is another way of writing fg(x): the output of g is fed directly into f. The method is to replace every x in f with the entire expression g(x), so with f(x) = 2x + 1 and g(x) = x², f(g(x)) = 2x² + 1.
What are common mistakes with composite functions?
The two common mistakes with composite functions are applying the functions in the wrong order and making an algebraic slip when substituting an expression rather than a number into a function. The rule that the function closest to x acts first avoids the first error.